We have calculated the self-diffusion constant in monodisperse linear polymer melts using a recent analytical theory of stress relaxation that incorporates contour-length fluctuations as well as reptation. We find over a broad range of chain lengths N that D similar to N-2.25 before crossing over to the asymptotic reptation result of D similar to N-2. This result is in agreement with recent experiments and single-chain simulations and resolves a long-standing discrepancy between the scaling of the zero-shear viscosity and the diffusion constant.